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## Probability
There is essentially one set of mathematical rules for manipulating probability; these rules are listed under "Formalization of probability" below. (There are other rules for quantifying uncertainty, such as the Dempster-Shafer theory and fuzzy logic, but those are essentially different and not compatible with the laws of probability as they are usually understood.) However, there is ongoing debate over what, exactly, the rules apply to; this is the topic of probability interpretations. The general idea of probability is often divided into two related concepts: - Aleatory probability, which represents the likelihood of future events whose occurrence is governed by some
*random*physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information, and phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel, and an example of the second kind is radioactive decay. - Epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are:
- Assign a probability to the proposition that a proposed law of physics is true.
- Determine how "probable" it is that a suspect committed a crime, based on the evidence presented.
## Historical remarks## Formalization of probabilityLike other theories, the theory of probability is a representation of probabilistic concepts in formal terms -- that is, in terms which are devoid of meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain. There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation, sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's formulation, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details:
- a probability is a number between 0 and 1;
- the probability of an event or proposition and its complement must add up to 1; and
- the joint probability of two events or propositions is the product of the probability of one of them and the probability of the second, conditional on the first.
## Representation and interpretation of probability valuesProbabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events. The odds of heads-up, for the tossed coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".
Odds There remains the question of exactly what can be assigned probability, and how the numbers so assigned can be used; this is the question of probability interpretations. There are some who claim that probability can be assigned to any kind of an uncertain logical proposition; this is the Bayesian interpretation. There are others who argue that probability is properly applied only to propositions concerning sequences of repeated experiments or sampling from a large population; this is the frequentist interpretation. There are several other interpretations which are variations on one or the other of those, or which have less acceptance at present. ## DistributionsThere are several equivalent ways to specify a probability distribution. Perhaps the most common is to specify a probability density function. Then the probability of an event or proposition is obtained by integrating the density function. The distribution function may also be specified directly. In one dimension, the distribution function is called the cumulative distribution function. Probability distributions can also be specified via moments or the characteristic function, or in still other ways.
A distribution is called a Important discrete distributions include the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution. Important continuous distributions include the normal distribution, the gamma distribution, the Student's t-distribution, and the exponential distribution. ## Probability in mathematicsProbability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem.
To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor - certainly, while we might
One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent - that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform
As
a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,
eventually the number of heads over the number of total flips will become arbitraily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.
The ## Remarks on probability calculationsThe difficulty of probability calculations lie in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events. Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem demonstrates the pitfalls nicely. To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem that explains the use of conditional probabilities in case where the occurrence of two events is related. ## Applications of probability theory to everday lifeA major impact of probability theory on everyday life is in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable impact on the population as a whole, statistically. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the impact of such choices, which makes probability measures a political matter. A good example is the impact of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the impact of such groupthink on pricing, on policy, and on peace and conflict. It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound impact on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy. ## See also- Bayesian probability
- Bernoulli process
- Games of chance
- Game theory
- Decision theory
- Cox's theorem
- Information theory
- Law of averages
- Law of large numbers
- Normal distribution
- Random variable
- Statistics
- List of statistical topics
- Stochastic process
- Wiener process
## External links
- Edwin Thompson Jaynes.
*Probability Theory: The Logic of Science*. Preprint: Washington University, (1996). -- http://omega.albany.edu:8008/JaynesBook.html and http://bayes.wustl.edu/etj/prob/book.pdf - Probabilistic football prediction competition, probabilistic scoring and further reading.
## Quotations- Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
- Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge."
*Théorie Analytique des Probabilités*, 1812. - Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace).
*Probability, Statistics, and Truth,*p 9. Dover edition, 1981 (republication of second English edition, 1957).
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